A Distant PlanetaryMass Solar Companion May Have Produced Distant. Detached Objects


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1 A Distant PlanetaryMass Solar Companion May Have Produced Distant Detached Objects Rodney S. Gomes a John J. Matese b Jack J. Lissauer c a Observatório Nacional, Rua General José Cristino, 77, , Rio de Janeiro, RJ, Brazil b Department of Physics, University of Louisiana at Lafayette, Lafayette, LA, c Space Science and Astrobiology Division, MS 2453, NASA Ames Research Center, Moffett Field, CA, Pages: 41 including 16 Figures Submitted to the journal ICARUS on 16 December 2005
2 Proposed Running Head: Producing Distant Detached Objects Editorial Correspondence: John J. Matese Department of Physics University of Louisiana at Lafayette Lafayette, LA, Tel: (337) Fax: (337)
3 ABSTRACT Most known transneptunian objects (TNO s) are either on low eccentricity orbits or could have been perturbed to their current trajectories via gravitational interactions with known bodies. However, one or two recentlydiscovered TNO s are distant detached objects (DDO s) (perihelion, q, > 40 AU and semimajor axis, a, > 50 AU) whose origins are not as easily understood. We investigate the parameter space of a hypothetical distant planetarymass solar companion which could detach the perihelion of a Neptunedominated TNO into a DDO orbit. Perturbations of the giant planets are also included. The problem is analyzed using two models. In the first model, we start with a distribution of undetached, lowinclination TNO s having a wide range of semimajor axes. The planetary perturbations and the companion perturbation are treated in the adiabatic, secularly averaged tidal approximation. This provides a starting point for a more detailed analysis by providing insights as to the companion parameter space likely to create DDO s. The second model includes the companion and the planets and numerically integrates perturbations on a sampling that is based on the real population of scattered disk objects (SDO s). A single calculation is performed including the mutual interactions and migration of the planets. By comparing these models, we distinguish the distant detached population that can be attributable to the secular interaction from those that require additional planetary perturbations. We find that a DDO can be produced by a hypothetical Neptunemass companion having semiminor axis, b c, 2000 AU or a Jupitermass companion with b c 5000 AU. DDO s produced by such a companion are likely to have small inclinations to the ecliptic only if the companion s orbit is significantly inclined. We also discuss the possibility that the tilt of the planets invariable plane relative to the solar equatorial plane has been produced by such a hypothetical distant planetarymass companion. Perturbations of a companion on Oort cloud comets are also considered. Key Words: TransNeptunian Belt, Oort Cloud, comets2003 VB 12, 2000 CR 105 3
4 1. Introduction Nearly 1000 minor planets orbiting the Sun on paths that are primarily or entirely beyond the orbit of Neptune have received official designations from the Minor Planet Center. Almost all of these transneptunian objects (TNO s) either have low eccentricity orbits with semimajor axes a < 50 AU, are trapped in meanmotion orbital resonances with Neptune, or travel on orbits that at least at times drop down close enough to Neptune to have their dynamics controlled by gravitational interactions with Neptune or other giant planets. In contrast, two objects, 2000 CR 105 (q = 44 AU, a = 221 AU, i = 22.7 ; discovered in the Deep Ecliptic Survey) and Sedna (2003 VB 12 ; q = 76 AU, a = 489 AU, i = 11.9 ; Brown, Trujillo and Rabinowicz (2004)), travel on eccentric orbits with sufficiently distant (detached) perihelia that they cannot have been perturbed onto their current trajectories by the actions of the planets as currently configured (Emel yanenko, Asher and Bailey (2003)). Similarly, the aphelia of Sedna and 2000 CR 105 are too small for the galactic tides to have detached them from the influence of the known planets. In this sense, they appear to be the first discovered true inner Oort cloud objects. Gomes, et al. (2005a), discuss resonant mechanisms for converting objects in the scattered disk (SD, defined as having semimajor axis exceeding 50 AU) into distant detached objects (DDO s). They find that resonance capture during the epoch of giant planet migration can produce DDO s such as 2000 CR 105, but the processes that they simulated cannot produce DDO s with a > 260 AU. Moreover, Sedna s inclination is too low for its orbit to be produced by the combination of a mean motion resonance and the Kozai mechanism with Neptune. Morbidelli and Levison (2004) consider, and reject, three additional mechanisms, (i) the passage of Neptune through a higheccentricity stage, (ii) the past existence of massive planetary embryos in the Kuiper belt or scattered disk, and (iii) tidal perturbations on scattered disk objects exerted by a hypothetical massive transneptunian disk at early epochs. The only options which they find to give satisfactory results are the passage of a lowvelocity solarmass star at about 800 AU, or the capture of extrasolar planetesimals from a lowmass star or brown dwarf encountering the Sun at low velocity. Such close, slow encounters are only plausible when the Sun was young and surrounded by siblings born from 4
5 the same molecular cloud. Morbidelli and Levison (2004) note that creating a population of extended scattered disk objects in which the largest orbits also have the largest perihelia requires a perturbation from the outside, but they do not discuss the possibility that the external perturbation could come from a distant planetarymass solar companion. We consider that option here. Our goal is to delineate the possible parameter space of a hypothetical solar companion which would be capable of detaching the orbit of an SDO from the dominance of Neptune. Section 2 describes an investigation of this problem using the adiabatic secular analysis and presents the ranges of companion parameters that could produce a Sednalike DDO in this approximation. Direct numerical integrations including the four major planets and the hypothetical solar companion are then presented in Section 3. In Section 4, we discuss the possibility that a companion has produced the observed misalignment between the invariable plane and the Sun s equatorial plane and also consider the effects that a companion would have on Oort cloud comets. We compare the results for secular and full perturbations, summarize and give our conclusions in Section 5. 5
6 2. Perturbing Scattered Disk Objects: The Secular Analysis The equations of motion of the eccentricity vector and the angular momentum vector (see Fig. 1) of an SDO have been analyzed by Matese, Whitmire and Lissauer (2006). The tidal perturbations of a hypothetical distant planetarymass solar companion (M c, r c ), and the known planets (M p, r p ), treated as massive rings, are included in a secularly averaged manner, with possible resonances and impulsive interactions being ignored. The inferences discussed in this section should be used only as a guide for a more detailed analysis, contained in the next section, in which these effects are not ignored. We approximate the companion motion as an invariant ellipse of mass and orbital parameters M c, a c, e c, i c having orbit normal ˆn c. The heliocentric SDO position is denoted by r; the scaled angular momentum vector, h, and the eccentricity vector, e, are given by h r ṙ µa, e ṙ h µ/a ˆr, (1) where µ GM G(M + p M p) includes the contributions from the known planets as well as the Sun. Sequentially performing secular averages over the short (SDO) orbital period and the long (companion) orbital period, in the tidal limits we obtain ḣ = 2(ˆn p h) ˆn p h h 5 τ p + 5(ˆn c e) ˆn c e (ˆn c h) ˆn c h τ c (2) ė = (h2 3(ˆn p h) 2 ) h e 2(ˆn p h) (ˆn p (h e))h h 7 τ p + (3) + h e + 4(ˆn c e) ˆn c h + (ˆn c (h e))ˆn c τ c where ˆn p is the normal direction to the ecliptic plane with 1 3 p µ prp 2 τ p 8 and µa 7 1 3M c µa 3 τ c 4Mb 3 c (4) 6
7 n p (ecliptic plane normal) n c (companion plane normal) e (DDO eccentricity) h (DDO angular momentum) i c i ω Ω (DDO ascending node) companion ascending node Fig. 1. Orientation definitions of the orbital vectors of the hypothetical planetarymass companion and the distant detached object. 7
8 where b c a c 1 e 2 c is the semiminor axis of the companion. These analytic forms are obtained using Mathematica (Wolfram Research (2003)). The largest planetary perturbation is from Neptune, which contributes 41% to the total. We see that the secular planetary interaction produces orbit normal precession around ˆn p, while a similar term in the secular companion interaction produces orbit normal precession around ˆn c. It is the term proportional to (ˆn c e) ˆn c e that serves to change q for largeeccentricity SDO s. Note that this term vanishes if the SDO and the companion are coplanar. The analysis reproduces the wellknown result that in the secular approximation planetary perturbations alone do not change e (Goldreich (1965)). The secularly averaged equations depend on the companion elements through the quantities ˆn c and τ c. There are several symmetries evident in the equations, such as their invariance when ˆn c ˆn c, i.e., i c π i c, and their independence of the companion perihelion direction, ê c. Orienting the coordinate axes as shown in Fig. 1, we see that the companion can be characterized by two parameters, i c and ρ c M c b 3, (5) c a strength parameter, both assumed to be constant here. The DDO orbit is characterized by a secularly constant semimajor axis, a, and four variable elements i, ω, Ω and e. The six coupled equations for the components of e and h are restricted by the two conserved quantities, h e = 0 and h 2 + e 2 = 1, which serve as checks on our numerical solutions. Our goal here is to guide the numerical modelling described in the next section, and as such the analysis is not meant to provide meaningful detailed predictions of the actual distributions. In that context, we assume that each possible SDO progenitor orbit of an DDO object was dominated by Neptune with initial values q = AU, i = These initial values are randomly sampled in the indicated ranges, with a randomly chosen between q and 1000 AU. The remaining two initial elements of the token orbits, Ω, ω, are randomly sampled over their ranges We then integrate the equations over an interval of years and record a, q and i at that time. The distributions are essentially unchanged if 8
9 integration times are extended beyond years. In Fig. 2, we show the results for a single value of ρ c = M AU 3. For example, this would describe the perturbation of a Neptunemass companion, M c = M, in an orbit with b c = 1500 AU, or equivalently a Jovianmass companion, M c = 10 3 M, in an orbit with b c = 4000 AU. We see that the ability of the secular companion perturbation to detach perihelia of an SDO is significantly reduced when a is small. It also shows that the DDO maintains the modest (initial) inclinations of the SDO only if the companion is itself significantly inclined to the ecliptic. The general features of Fig. 2 hold for other values of ρ c with a single exception, the value of a for which the onset of detachment occurs. We can estimate this critical value as follows. In Eq. (2), we compare the timescale for planetaryinduced precession with that of the companion perturbation term causing detachment. Ignoring the angular factors, for nearparabolic SDO s the ratio is strongly dependent on a: 5e 2 /τ c 2/(h 3 τ p ) 10 q 3/2 a 7/2 2 ρ c p M 2 pr. (6) p Detachment occurs when this ratio is O(1). Inserting planetary parameters and an initial value of q 35 AU and setting the ratio equal to 1 yields an estimate of the value of a for which the onset of detachment occurs, a detach 490 AU for the parameters used in Fig. 2. This is in reasonable agreement with the numerical results presented in Fig. 2. More generally, the estimate is a detach 550 AU (ρ c /10 14 M AU 3 ) 2/7 (7) in the secular approximation. Specifically, we find that detaching to perihelia > 75 AU at Sedna s semimajor axis requires M c > M AU 3 b c 3. (8) A distant companion orbit is subject to perturbations from passing stars and the galactic tide. Therefore, these parameters essentially describe the epoch when companion interactions with DDO s are strongest, i.e., when ρ c is largest. The galactic tide affects e c and i c, but changes are small for a c < 10,000 AU. In years osculations proceed through one halfcycle when a c 20,000 AU. 9
10 ic 0, perihelion AU ic 45, ic semimajor axis AU Fig. 2. The effects of secular perturbations, integrated over years, of scattered disk objects for the companion strength parameter ρ c M c /b 3 c = M AU 3 and various companion inclinations i c. The SDO s have perihelia between 32 AU and 38 AU at the beginning of the integrations. Equivalent example parameter sets are b c a c 1 e 2 c = 1500 AU, M c = M and b c = 3000 AU, M c = M. Gray dots i < 15, black dots i > 15, triangles Sedna and 2000 CR
11 To illustrate the subtle interplay between secular and impulsive interactions, we show in Fig. 3 an example case of the orbital evolution of a single massless particle interacting with the four known giant planets and the companion, obtained by direct numerical integration as discussed in Section 3.1 below. The particle is initially in a nearcircular smallinclination orbit just beyond Neptune, and the companion has parameters a c = 1500 AU, e c = 0, i c = 90, M c = M and ρ c = M c /b 3 c = M AU 3. During the initial stages of the 10 9 year integration, impulses by Neptune slowly pump the semimajor axis to 450 AU at which time companion secular effects are strong enough to detach it from Neptune dominance. Detachment occurs at a value of a that is in reasonable agreement with the analytic estimate for a detach given in Eq. 7. The secular interaction can return the particle to Neptune dominance, as shown by Matese, Whitmire and Lissauer (2006). This leads to a second brief impulsively dominated epoch and eventually the particle is secularly detached for a second time when a 500 AU. These impulsive effects are ignored in the secular analysis illustrated in Fig. 2 and therefore a secular approximation cannot produce a complete description of distributions. We can qualitatively understand the competing processes by considering the adependence of the energypumping and perihelion migration timescales during the impulsively dominated epoch da 1 dt a 3/2 and dq dt a h τ c a 2. (9) The illustration shown in Fig. 3 indicates that when a 400 AU, q can grow sufficiently rapidly that energy pumping effectively ceases, i.e., detachment occurs. Both detachment and reattachment occur when q 40 AU. 3. Perturbing Scattered Disk Objects: Direct Numerical Integrations Two models have been used for direct numerical integrations. The first model considers all four major planets (at their present orbits) and the companion as perturbers. The particles are massless and are started near Neptune so as to get promptly scattered. There are initially 1000 particles; 500 particles are 11
12 time Fig. 3. An example of the orbital evolution of a single massless particle interacting with the four known giant planets and the companion, obtained by direct numerical integration. The particle is initially in a nearcircular smallinclination orbit just beyond Neptune and the companion has parameters a c = 1500 AU, e c = 0, i c = 90, M c = M and ρ c = M AU 3. 12
13 started uniformly with a from 27 to 29 AU; 500 particles are started uniformly with a from 31 to 33 AU. The initial eccentricities are randomly distributed between 0 and 0.1 and the initial inclinations randomly distributed between 0 and 1. The second model includes the four planets started in compact orbits and a disk of massive particles just outside the outermost planet; only a single run was performed for this computationallydemanding model. The particles perturb the planets and companion and these perturb all objects, but the particles do not affect one another. This model induces a planetary migration in an abrupt sense as described in Gomes, et al. (2005b), Tsiganis, et al. (2005) and Morbidelli, et al. (2005) pointing towards an explanation of the lunar heavy bombardment, the origin of the Trojans and the present orbital structure of the giant planets. We also considered the galactic tide in this run. The set of runs for the first model used 1 year for the steplength. For the migration model 0.5 year was used in the beginning and 1 year after the LHB had calmed down and the planets were close to their present orbits. All integrations were performed using the MERCURY package (Chambers (1999)) Four Known Planets and Companion For this model (massless particles), we begin by showing examples meant to compare with the semianalytical approach of Section 2. First we compare the distribution created by two different companions. Both components have circular orbits and their semimajor axes and masses are, respectively, a c = 1500 AU, M c = M and a c = 3000 AU, M c = M. These two sets of companion s parameters correspond to the same strength parameter, ρ c, (Eq. 5) and should induce the same a q distribution for a common companion inclination if the secular approximation is appropriate. Each numerical integration simulated 10 9 years. Fig. 4 shows the pair of distributions for i c = 90. The inner border of the detached regions and the distributions of low inclination objects are quite similar, suggesting that the secular effects as developed in Section 2 play a prominent role in the establishment of the a q distribution of detached scattered objects. Differences with the direct numerical integration model are of 13
14 Fig. 4. Direct numerical integration were undertaken with all major planets and a companion with ρ c = M AU 3 as perturbers. Distributions of semimajor axes and perihelia of particles started near Neptune after 10 9 years of evolution are plotted. Upper panel M c = M, a c = 1500 AU, e c = 0, i c = 90 ; lower panel M c = M, a c = 3000 AU, e c = 0, i c = 90. Gray dots i < 15, black dots i > 15, triangles Sedna and 2000 CR 105. These plots can be compared to the secular analysis results given in the bottom panel of Fig
15 Fig. 5. At 10 9 years, the distributions of semimajor axes and perihelia of particles started near Neptune. Direct numerical integration were undertaken with all major planets and a companion with ρ c = M AU 3 as perturbers. Companion parameters M c = M, a c = 1500 AU, e c = 0; inclinations as indicated in each panel. Gray dots i < 15, black dots i > 15, triangles Sedna and 2000 CR 105. These plots can be compared to the secular analysis results given in Fig
16 two kinds. First, the larger, more distant companion is more effective at detaching perihelia of scattered objects with larger semimajor axes. Second, the smaller, closer companion forms a 28% more massive inner Oort cloud at 10 9 years. The reason for this deserves a deeper investigation. A larger M c raises the perihelia of scattered objects in a shorter timescale, but also brings them back to Neptunecrossing orbits in the same short timescale. Moreover, the companion can directly scatter the objects away from the inner Oort cloud. There must be an ideal companion (mass, distance from the Sun, eccentricity and inclination) that will create the largest inner Oort cloud population at the Solar System s age, but this goes beyond the scope of this paper. Note also that even for this high value of i c there is no significant Kozai mechanism altering the companion s orbit; e c only goes up to 0.05 and i c varies by only ±2. For the smaller, closer, companion, integrations with i c = 0, 45, 135, 180 were also undertaken. The a q distributions created by this companion are shown in Fig. 5. The inclinations equidistant from 90 are placed side by side so as to compare their similarities. In principle, by the secular theory these distributions should be equivalent, which is fairly well confirmed by comparing Figs. 4 and 5 with Fig. 2. We conclude that secular analysis fails where it is expected to fail, i.e., when the tidal approximation is invalid, particularly when impulses can occur. Nonetheless, it provides a useful approach to delineating the parameter space that needs to be numerically studied in investigating the possibility that the Distant Detached Population (DDP) was produced by a hypothetical distant planetary mass companion. Now we give three other examples covering a wide range of masses and semimajor axes for the companion. High eccentricities are used in these cases. In one extreme, we consider a companion with a c = 5000 AU, e c = 0.9, i c = 70 and M c = M. Figure 6 shows the a q distribution at one billion years of integration. This companion has a ρ c = M AU 3, thus much higher than those given in the examples above and also higher than the value needed to detach Sedna and 2000 CR 105 (see Eq. 7). In fact, Fig. 6 shows that DDO s are produced not only at Sedna s semimajor axis but also at 2000 CR 105 semimajor axis. DDO s are indeed produced for semimajor axes just above 100 AU. This example is instructive as these parameters for the companion also induce a 6 tilt of the invariable plane at 16
17 Fig. 6. Distributions of semimajor axes and perihelia of particles started near Neptune. Direct numerical integration were undertaken with all major planets and a companion with ρ c = M AU 3 and i c = 70 as perturbers. Companion parameters M c = M, a c = 5000 AU, e c = 0.9. Gray dots i < 15, black dots i > 15, triangles Sedna and 2000 CR
18 Fig. 7. Same as Fig. 6, except that the companion parameters M c = M, a c = 1000 AU, e c = 0.94, i c = 5, and thus ρ c = M AU 3. 18
19 Fig. 8. Same as Figs. 6 and 7, except for a companion parameters M c = 10 4 M, a c = 1500 AU, e c = 0.4, i c = 90 and thus ρ c = M AU 3. 19
20 the age of the Solar System (see Section 4.1). The second case examines another extreme with an Earth mass planet (M c = M ), a c = 1000 AU, e c = 0.94, i c = 5, that corresponds to ρ c = M AU 3, again large enough to detach Sedna but not 2000 CR 105. Fig. 7 shows the a q distribution induced by this planet at 2 billion years of integration. Sedna is inside the IOC created by this Earth mass planet, but a 2000 CR 105 orbit is hardly detached by this planet. In this case the secular theory is inappropriate since the companion orbit is partially interior to DDO orbits. Secular theory fails in that it predicts that detached orbits should have large inclinations, the opposite of that obtained in the numerical analysis. The third eccentric example is for a companion with a c = 1500 AU, e c = 0.4, i c = 90 and M c = 10 4 M. These parameters correspond to a ρ c = M AU 3, which is large enough to detach Sedna but not 2000 CR 105 according to Eq. 7. The a q distribution at 2 billion years of integration is shown in Fig. 8. We see Sedna well inside an IOC whose members have semimajor axes as low as 200 AU. This also includes 2000 CR 105, a more optimistic result than predicted by the secular theory. As Fig. 5 suggests, 90 companions create IOC s that extend to lower semimajor axes Four Migrating Planets, Companion and Galactic Tide One single integration was done with a fairly complete evolutionary model. Here we suppose that the giant planets had primordially more compact orbits and there was a disk of planetesimals just outside the outermost planet. The initial semimajor axes for the planets are 5.45, 8.18, 11.5 and 14.2 AU. The disk extends from 15.5 AU to 34 AU with a surface density proportional to r 1. It is composed of 10,000 equally massive particles totalling 35 M. Both the planets and the particles initial eccentricities and inclinations are small. A solar companion with parameters a c = 1500 AU, e c = 0.4, i c = 40 and M c = 10 4 M (as in the final example presented in Section 3.1) as well as the galactic tidal perturbation are also included from the beginning of the integration. We used the same model for the galactic tide as that in Dones, et al. (2005) and references therein. In this model, the accelerations in the directions defined by the galactic 20
21 Fig. 9. Distributions of semimajor axes and perihelia of particles started in a disk just outside the giant planets in initial compact orbits. Direct numerical integrations are done with the major planets and a companion as perturbers. The particles have mass and perturb the planets and companion (but not one another), inducing planetary migration. The galactic tide was also included in this integration. The results are after years, when the planets are near their present position. Companion parameters M c = 10 4 M, a c = 1500 AU, e c = 0.4, ρ c = M AU 3. Gray dots i < 15, black dots i > 15, triangles Sedna and 2000 CR
22 Fig. 10. Same as Fig. 9 in a larger scale that also shows particles trapped in the outer Oort cloud. Vertical and horizontal lines define each population, according to: inner Oort cloud a < 10, 000 AU, q > 60 AU outer Oort cloud a > 10, 000 AU, q > 60 AU scattered population q < 60 AU. 22
23 Fig. 11. Evolution of total mass trapped in the inner Oort cloud, outer Oort cloud and scattered populations, according to the same definitions used for Fig
24 frame { x, ỹ, z} are: F x = Ω 2 0 x, Fỹ = Ω 2 0 ỹ, F z = 4π G ρ 0 z, (10) with Ω 0 = 26 km/s/kpc and ρ 0 = 0.1 M /pc 3. Planetary migration is induced by the massive disk of planetesimals, whose parameters are chosen so that the planets stop near their present positions after scattering the planetesimals inwards and outwards according to a late heavy bombardment scenario (Gomes, et al. (2005b), Tsiganis, et al. (2005) and Morbidelli, et al. (2005)). Fig. 9 shows the planetesimals a q distribution obtained at Solar System age. Both Sedna and 2000 CR 105 are shown in the a q distribution induced by the companion. This single example, with ρ c = M AU 3, is instructive since it also estimates the total mass left in the inner Oort cloud. Fig. 10 presents the a q distribution in another scale showing the distribution in the inner Oort cloud (induced by the companion), in the outer Oort cloud (modified by the galactic tide) and in the scattered population. Figure 11 shows the time evolution of the mass in each population. The limits for each population are defined as: inner Oort cloud: semimajor axis below 10,000 AU and perihelion above 60 AU; outer Oort cloud: semimajor axis above 10,000 AU; scattered population: semimajor axis below 10,000 AU and perihelion below 60 AU. With these definitions, we have at Solar System age: 1.07 M for the inner Oort cloud, 1.34 M for the outer Oort cloud and 0.14 M for the scattered population. These numbers yield a ratio of the mass in the scattered population to the mass in the outer Oort cloud of 10%. In comparison with the total Oort cloud mass, this ratio drops to 6%. If we consider the more usual definition of the scattered population as q < 45 AU, then this ratio drops to 4%. These ratios are much higher than observational estimates, which yield a ratio of the scattered population mass to the Oort cloud mass no bigger than 0.5% (Dones, et al. (2005), Duncan, et al. (2004)). The presence of a companion does not seem to change significantly these mass ratios as compared to simulations with no companion (Dones, et al. (2005)). On the other hand, cometary mass estimates from observations are based on the number of bodies in each population that succeeds in appearing as an observable comet, and the dynamics that 24
25 rules the transfer of icy bodies in each cometary population to the observable zone also depends on the companion s perturbation. This should be taken into account to estimate the mass in each population based on observable comets. As an example, a scattered object with semimajor axis in the companion s influence zone may not follow its way from that disk to a JFC due to the companion s perturbation. For instance, it may have its perihelion increased once more. Before a deeper investigation on this issue is developed, one should not draw strong conclusions about the validity of the observed mass ratios in cometary populations in the presence of the perturbation from a solar companion. Fernández and Brunini (2000) consider various scenarios for a dense primordial environment of the Solar System. They find that a tightly bound inner Oort cloud containing 2 M can be formed at a timescale of a few million years. This is an amount of mass comparable to that shown in the complete Oort cloud in Fig. 11. The inclination distribution produced depends on the details of their modelling. Assessing the consequences of these populations in the influx of comets into the inner Solar System requires additional simulations to obtain statistically robust results and thus are not investigated in this paper. However, it must be noted that the inner Oort cloud created by the companion model is live, in contrast to the fossil populations created by passing star and dense early environment models. This means that the inner Oort cloud can continually replenish both the scattered population and the outer Oort cloud. Fig. 12 shows the distribution of semimajor axes with perihelia for q < 10 AU, taken from the last 100 million years of the migration model. We notice an enhancement of low q objects for a 10,000 AU and above, what is about expected from a single Oort cloud model, and an enhancement for a < 2000 AU, but for q > 5 AU. This figure seems to indicate that particles that attain small q by the companion s secular perturbation eventually get their semimajor axes decreased, thus feeding the Centaur population and eventually the Halley and Jupiter family comets populations. This point however needs a deeper investigation that our data are not able to confirm. This figure also suggests that the companion is not capable of sending a significant number of icy objects directly to the inner Solar System. Nonetheless, of all the particles whose q was detected lower than 10 AU during the last 100 million years of integration, about 25
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